Contraction Groups and Passage to Subgroups and Quotients for Endomorphisms of Totally Disconnected Locally Compact Groups

TL;DR

This paper extends fundamental concepts like scale, tidy subgroups, and contraction groups from automorphisms to endomorphisms of totally disconnected locally compact groups, including their quotients and contraction behaviors.

Contribution

It generalizes key structural results from automorphisms to endomorphisms, enhancing understanding of group dynamics and subgroup behavior in this broader context.

Findings

Extension of scale and tidy subgroup concepts to endomorphisms

Results on contraction groups and quotients for endomorphisms

Analysis of attraction domains around invariant subgroups

Abstract

The concepts of the scale and tidy subgroups for an automorphism of a totally disconnected locally compact group were defined in seminal work by George A. Willis in the 1990s, and recently generalized to the case of endomorphisms (G. A. Willis, Math. Ann. 361 (2015), 403--442). We show that central facts concerning the scale, tidy subgroups, quotients, and contraction groups of automorphisms extend to the case of endomorphisms. In particular, we obtain results concerning the domain of attraction around an invariant closed subgroup.